Half-life of radioactive elements - what is it and how is it determined? Half-life formula. Radioactivity
Lecture 2. The basic law of radioactive decay and the activity of radionuclides
The rate of decay of radionuclides is different - some decay faster, others slower. Speed indicator radioactive decay is radioactive decay constant, λ [sec-1], which characterizes the probability of the decay of one atom in one second. For each radionuclide, the decay constant has its own value; the larger it is, the faster the nuclei of the substance decay.
The number of decays recorded in a radioactive sample per unit time is called activity (a ), or the radioactivity of the sample. The activity value is directly proportional to the number of atoms N radioactive substance:
a =λ· N , (3.2.1)
Where λ – radioactive decay constant, [sec-1].
Currently, according to the current International System of Units SI, the unit of measurement of radioactivity is becquerel [Bk]. This unit received its name in honor of the French scientist Henri Becquerel, who discovered the phenomenon of natural radioactivity of uranium in 1856. One becquerel equals one decay per second 1 Bk = 1 .
However, the non-system unit of activity is still often used – curie [Ki], introduced by the Curies as a measure of the decay rate of one gram of radium (in which ~3.7 1010 decays occur per second), therefore
1 Ki= 3.7·1010 Bk.
This unit is convenient for assessing activity large quantities radionuclides.
The decrease in radionuclide concentration over time as a result of decay obeys an exponential relationship:
, (3.2.2)
Where N t– the number of atoms of a radioactive element remaining after time t after the start of observation; N 0 – number of atoms at the initial moment of time ( t =0 ); λ – radioactive decay constant.
The described dependence is called basic law of radioactive decay .
The time it takes for half of the total number radionuclides is called half-life, T½ . After one half-life, out of 100 radionuclide atoms, only 50 remain (Fig. 2.1). Over the next similar period, of these 50 atoms, only 25 remain, and so on.
The relationship between half-life and decay constant is derived from the equation of the fundamental law of radioactive decay:
at t=T½ And
we get https://pandia.ru/text/80/150/images/image006_47.gif" width="67" height="41 src="> Þ ;
https://pandia.ru/text/80/150/images/image009_37.gif" width="76" height="21">;
i.e..gif" width="81" height="41 src=">.
Therefore, the law of radioactive decay can be written as follows:
https://pandia.ru/text/80/150/images/image013_21.gif" width="89" height="39 src=">, (3.2.4)
Where at – drug activity over time t ; a0 – activity of the drug at the initial moment of observation.
It is often necessary to determine the activity of a given amount of any radioactive substance.
Remember that the unit of quantity of a substance is the mole. A mole is the amount of a substance containing the same number of atoms as are contained in 0.012 kg = 12 g of the carbon isotope 12C.
One mole of any substance contains Avogadro's number N.A. atoms:
N.A. = 6.02·1023 atoms.
For simple substances(elements) the mass of one mole corresponds numerically to atomic mass A element
1mol = A G.
For example: For magnesium: 1 mol 24Mg = 24 g.
For 226Ra: 1 mol 226Ra = 226 g, etc.
Taking into account what has been said in m grams of the substance will be N atoms:
https://pandia.ru/text/80/150/images/image015_20.gif" width="156" height="43 src="> (3.2.6)
Example: Let's calculate the activity of 1 gram of 226Ra, which has λ = 1.38·10-11 sec-1.
a= 1.38·10-11·1/226·6.02·1023 = 3.66·1010 Bq.
If a radioactive element is included in the composition chemical compound, then when determining the activity of a drug, it is necessary to take into account its formula. Taking into account the composition of the substance, it is determined mass fraction χ radionuclide in a substance, which is determined by the ratio:
https://pandia.ru/text/80/150/images/image017_17.gif" width="118" height="41 src=">
Example of problem solution
Condition:
Activity A0 radioactive element 32P per day of observation is 1000 Bk. Determine the activity and number of atoms of this element after a week. Half life T½ 32P = 14.3 days.
Solution:
a) Let’s find the activity of phosphorus-32 after 7 days:
https://pandia.ru/text/80/150/images/image019_16.gif" width="57" height="41 src=">
Answer: after a week, the activity of the drug 32P will be 712 Bk, and the number of atoms of the radioactive isotope 32P is 127.14·106 atoms.
Security questions
1) What is the activity of a radionuclide?
2) Name the units of radioactivity and the relationship between them.
3) What is the radioactive decay constant?
4) Define the basic law of radioactive decay.
5) What is half-life?
6) What is the relationship between activity and mass of a radionuclide? Write the formula.
Tasks
1. Calculate activity 1 G 226Ra. T½ = 1602 years.
2. Calculate activity 1 G 60Co. T½ = 5.3 years.
3. One M-47 tank shell contains 4.3 kg 238U. Т½ = 2.5·109 years. Determine the activity of the projectile.
4. Calculate the activity of 137Cs after 10 years, if at the initial moment of observation it is equal to 1000 Bk. T½ = 30 years.
5. Calculate the activity of 90Sr a year ago if it is currently equal to 500 Bk. T½ = 29 years.
6. What kind of activity will 1 create? kg radioisotope 131I, T½ = 8.1 days?
7. Using reference data, determine activity 1 G 238U. Т½ = 2.5·109 years.
Using reference data, determine activity 1 G 232Th, Т½ = 1.4·1010 years.
8. Calculate the activity of the compound: 239Pu316O8.
9. Calculate the mass of a radionuclide with an activity of 1 Ki:
9.1. 131I, T1/2=8.1 days;
9.2. 90Sr, T1/2=29 years;
9.3. 137Cs, Т1/2=30 years;
9.4. 239Pu, Т1/2=2.4·104 years.
10. Determine mass 1 mCi radioactive carbon isotope 14C, T½ = 5560 years.
11. It is necessary to prepare a radioactive preparation of phosphorus 32P. After what period of time will 3% of the drug remain? Т½ = 14.29 days.
12. The natural potassium mixture contains 0.012% of the 40K radioactive isotope.
1) Determine the mass of natural potassium, which contains 1 Ki 40K. Т½ = 1.39·109 years = 4.4·1018 sec.
2) Calculate the radioactivity of the soil using 40K, if it is known that the potassium content in the soil sample is 14 kg/t.
13. How many half-lives are required for the initial activity of a radioisotope to decrease to 0.001%?
14. To determine the effect of 238U on plants, seeds were soaked in 100 ml solution UO2(NO3)2 6H2O, in which the mass of radioactive salt was 6 G. Determine the activity and specific activity of 238U in solution. Т½ = 4.5·109 years.
15. Identify Activity 1 grams 232Th, Т½ = 1.4·1010 years.
16. Determine mass 1 Ki 137Cs, Т1/2=30 years.
17. The ratio between the content of stable and radioactive isotopes of potassium in nature is a constant value. The 40K content is 0.01%. Calculate the radioactivity of the soil using 40K, if it is known that the potassium content in the soil sample is 14 kg/t.
18. Lithogenic radioactivity of the environment is formed mainly due to three main natural radionuclides: 40K, 238U, 232Th. The proportion of radioactive isotopes in the natural sum of isotopes is 0.01, 99.3, ~100, respectively. Calculate radioactivity 1 T soil, if it is known that the relative content of potassium in the soil sample is 13600 g/t, uranium – 1·10-4 g/t, thorium – 6·10-4 g/t.
19. In shells bivalves discovered 23200 Bq/kg 90Sr. Determine the activity of samples after 10, 30, 50, 100 years.
20. The main pollution of closed reservoirs in the Chernobyl zone took place in the first year after the accident at the nuclear power plant. In the bottom sediments of the lake. Azbuchin in 1999 discovered 137Cs with a specific activity of 1.1·10 Bq/m2. Determine the concentration (activity) of fallen 137Cs per m2 of bottom sediments as of 1986-1987. (12 years ago).
21. 241Am (T½ = 4.32·102 years) is formed from 241Pu (T½ = 14.4 years) and is an active geochemical migrant. Taking advantage reference materials, calculate with an accuracy of 1% the decrease in the activity of plutonium-241 over time, in which year after Chernobyl disaster 241Am's formation environment will be maximum.
22. Calculate the activity of 241Am in the emissions of the Chernobyl reactor as of April
2015, provided that in April 1986 the activity of 241Am was 3.82 1012 Bk,Т½ = 4.32·102 years.
23. 390 were found in soil samples nCi/kg 137Cs. Calculate the activity of samples after 10, 30, 50, 100 years.
24. Average concentration of lake bed pollution. Glubokoye, located in the Chernobyl exclusion zone, is 6.3 104 Bk 241Am and 7.4·104 238+239+240Pu per 1 m2. Calculate in what year these data were obtained.
It was formulated after Becquerel discovered the phenomenon of radioactivity in 1896. It consists in the unpredictable transition of one type of nuclei to another, while they release different particles of elements. The process can be natural when it manifests itself in isotopes existing in nature, and artificial, in cases where they are obtained in the nucleus that decays, is considered the mother, and the resulting one is considered the daughter. In other words, the basic law of radioactive decay involves the arbitrary natural process of one nucleus turning into another.
Becquerel's research showed the presence of previously unknown radiation in uranium salts, which affected the photographic plate, filled the air with ions and tended to pass through thin metal plates. The experiments of M. and P. Curie with radium and polonium confirmed the conclusion described above, and a new concept appeared in science, called the doctrine
This theory, reflecting the law of radioactive decay, is based on the assumption of a spontaneous process that obeys statistics. Since individual nuclei decay independently of each other, it is believed that, on average, the number of decayed ones over a certain period of time is proportional to those that have not decayed by the time the process ends. If you follow the exponential law, then the number of the latter decreases significantly.
The intensity of the phenomenon is characterized by two main properties of radiation: the so-called half-life and the calculated average life span of the radioactive nucleus. The first fluctuates between millionths of a second and billions of years. Scientists believe that such nuclei do not age, and for them there is no concept of age.
The law of radioactive decay is based on the so-called displacement rules, and they, in turn, are a consequence of the theory of conservation and mass number. It has been experimentally established that the action magnetic field acts in different ways: a) the deflection of rays occurs as positively charged particles; b) as negative; c) do not show any reaction. From this it follows that there are three types of radiation.
There are just as many varieties of the decay process itself: with the release of an electron; positron; absorption of one electron by the nucleus. It has been proven that nuclei whose structure corresponds to lead undergo decay with emission. The theory was called alpha decay and was formulated by G. in 1928. The second type was formulated in 1931 by E. Fermi. His research showed that instead of electrons, some types of nuclei emit opposite particles - positrons, and this is always accompanied by the emission of a particle with zero electrical charge and rest mass, neurino. The simplest example of beta decay is the transition of a neuron into a proton with a time period of 12 minutes.
These theories, which consider the laws of radioactive decay, were the main ones until 1940 of the 19th century, until Soviet physicists G.N. Flerov and K.A. Petrzhak discovered another type, during which uranium nuclei spontaneously split into two equal particles. In 1960, two-proton and two-neutron radioactivity was predicted. But to this day, this type of decay has not received experimental confirmation and has not been detected. Only proton radiation was discovered, in which a proton is ejected from the nucleus.
It is quite difficult to deal with all these issues, although the law of radioactive decay itself is simple. It's not easy to figure it out physical meaning and, of course, the presentation of this theory goes far beyond the physics curriculum as a subject at school.
Radioactive decay of nuclei of the same element occurs gradually and with at different speeds for different radioactive elements. It is impossible to specify in advance the moment of nuclear decay, but it is possible to establish the probability of the decay of one nucleus per unit time. The probability of decay is characterized by the coefficient "λ" - the decay constant, which depends only on the nature of the element.
Law of radioactive decay.(Slide 32)
It has been experimentally established that:
Over equal periods of time, the same proportion of available (i.e., not yet decayed at the beginning of a given interval) nuclei of a given element decays.
Differential form of the law of radioactive decay.(slide 33)
Establishes the dependence of the number of undecayed atoms in at the moment time from the initial number of atoms at the zero moment of reference, as well as from the decay time "t" and the decay constant "λ".
N t - available number of cores.
dN is the decrease in the available number of atoms;
dt - decay time.
dN ~ N t dt Þ dN = –λ N t dt
“λ” is the proportionality coefficient, the decay constant, characterizing the proportion of available nuclei that have not yet decayed;
“–” means that over time the number of decaying atoms decreases.
Corollary #1:(slide 34)
λ = –dN/N t dt - relative rate of radioactive decay for of this substance there is a constant value.
Corollary #2:
dN/N t = – λ · Nt - the absolute rate of radioactive decay is proportional to the number of undecayed nuclei at time dt. It is not "const", because will decrease over time.
4. Integral form of the law of radioactive decay.(slide 35)
Sets the dependence of the number of remaining atoms at a given time (N t) on their initial number (N o), time (t) and decay constant "λ". The integral form is obtained from the differential one:
1. Let's separate the variables:
2. Let’s integrate both sides of the equality:
3. Let's find the integrals 
Þ 
-general solution
4. Let's find a particular solution:
If t = t 0 = 0 Þ N t = N 0 , Let's substitute these conditions into the general solution
(start ( original number
decay) of atoms)
Þ
Thus:
integral form of the law r/act. disintegration
Nt - the number of undecayed atoms at the moment of time t ;
N 0 - initial number of atoms at t = 0 ;
λ - decay constant;
t - decay time
Conclusion: The available number of undecayed atoms is ~ the original quantity and decreases over time according to an exponential law. (slide 37)
Nt= N 0 2 λ 1 λ 2 >λ 1 Nt = N 0 e λ t
5. Half-life and its relationship with the decay constant. ( slide 38,39)
Half-life (T) is the time it takes for half the original number of radioactive nuclei to decay.
It characterizes the rate of decay of various elements.
Basic conditions for determining "T":
1. t = T - half-life.
2. 
- half of the original number of cores for "T".
The connection formula can be obtained if these conditions are substituted into the integral form of the law of radioactive decay
1. 
2. Let’s shorten “N 0”. Þ 
3. 
4. Let's potentiate.
Þ 
5. 
The half-lives of isotopes vary widely: (slide40)
238 U ® T = 4.51 10 9 years
60 Co ® T = 5.3 years
24 Na ® T = 15.06 hours
8 Li ® T = 0.84 s
6. Activity. Its types, units of measurement and quantification. Activity formula.(slide 41)
In practice, the main importance is the total number of decays occurring in a source of radioactive radiation per unit time => the measure of decay is determined quantitatively activity radioactive substance.
Activity (A) depends on the relative decay rate "λ" and on the available number of nuclei (i.e., on the mass of the isotope).
“A” characterizes the absolute decay rate of the isotope.
3 options for writing the activity formula: (slide 42,43)
I. From the law of radioactive decay in differential form follows:
Þ 
activity (absolute rate of radioactive decay).
activity
II. From the law of radioactive decay in integral form it follows:
1. 
(multiply both sides of the equality by “λ”).
Þ 
2. 
; 
(initial activity at t = 0)
3. 
The decrease in activity follows an exponential law
III. When using the formula for relating the decay constant "λ" to the half-life "T" it follows:
1. 
(multiply both sides of the equality by “ Nt
" to get activity). Þ 
and we get the formula for activity
2. 
Activity units:(slide 44)
A. System units of measurement.
A = dN/dt
1[disp/s] = 1[Bq] – becquerel
1Mdisp/s =10 6 disp/s = 1 [Rd] - rutherford
B. Non-system units of measurement.
[Ki] - curie(corresponds to the activity of 1g of radium).
1[Ci] = 3.7 10 10 [disp/s]- 1 g of radium decays in 1 s 3.7 10 10 radioactive nuclei.
Types of activity:(slide 45)
1. Specific is the activity per unit mass of a substance.
A beat = dA/dm [Bq/kg].
It is used to characterize powdery and gaseous substances.
2. Volumetric- is the activity per unit volume of a substance or medium.
A about = dA/dV [Bq/m 3 ]
It is used to characterize liquid substances.
In practice, the decrease in activity is measured using special radiometric instruments. For example, knowing the activity of the drug and the product formed during the decay of 1 nucleus, you can calculate how many particles of each type are emitted by the drug in 1 second.
If “n” neutrons are produced during nuclear fission, then a flux of “N” neutrons is emitted in 1 s. N = n A.
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Under radioactive decay, or just disintegration, understand the natural radioactive transformation of nuclei, which occurs spontaneously. An atomic nucleus undergoing radioactive decay is called maternal, the emerging core - subsidiaries.
The theory of radioactive decay is based on the assumption that radioactive decay is spontaneous process, subject to the laws of statistics. Since individual radioactive nuclei decay independently of each other, we can assume that the number of nuclei d N, decayed on average during the time interval from t to t + dt, proportional to the time period dt and number N undecayed nuclei at the time t:
where is a constant value for a given radioactive substance, called radioactive decay constant; The minus sign indicates that the total number of radioactive nuclei decreases during the decay process.
By separating the variables and integrating, i.e.
(256.2)
Where - seed number undecayed nuclei (at time t = 0), N- number of undecayed nuclei at a time t. Formula (256.2) expresses law of radioactive decay, according to which the number of undecayed nuclei decreases exponentially with time.
The intensity of the radioactive decay process is characterized by two quantities: the half-life and the average lifetime of the radioactive nucleus. Half life- the time during which the initial number of radioactive nuclei is halved on average. Then, according to (256.2),
Half-lives for naturally radioactive elements range from ten millionths of a second to many billions of years.
Total life expectancy dN cores is equal to 
. Having integrated this expression over all possible t(i.e. from 0 to ) and dividing by the initial number of cores, we get average life time radioactive nucleus:
(taken into account (256.2)). Thus, the average lifetime of a radioactive nucleus is the reciprocal of the radioactive decay constant.
Activity A nuclide (common name atomic nuclei that differ in the number of protons Z and neutrons N) in a radioactive source is the number of decays that occur with the nuclei of a sample in 1 s:
(256.3)
The SI unit of activity is becquerel(Bq): 1 Bq - activity of a nuclide, at which one decay event occurs in 1 s. To this day, nuclear physics also uses an off-system unit of activity of a nuclide in a radioactive source - curie(Ci): 1 Ci = 3.7×10 10 Bq. Radioactive decay occurs in accordance with the so-called displacement rules, allowing us to establish which nucleus arises as a result of the decay of a given parent nucleus. Offset rules:
For -decay
(256.4)
For -decay
(256.5)
where is the mother nucleus, Y is the symbol of the daughter nucleus, is the helium nucleus (-particle), is the symbolic designation of the electron (its charge is –1 and its mass number is zero). The displacement rules are nothing more than a consequence of two laws that apply during radioactive decays - the conservation of electric charge and the conservation of mass number: the sum of charges ( mass numbers) of the emerging nuclei and particles is equal to the charge (mass number) of the original nucleus.
Nuclei resulting from radioactive decay can, in turn, be radioactive. This leads to the emergence chains, or series, radioactive transformations ending with a stable element. The set of elements that form such a chain is called radioactive family.
From the displacement rules (256.4) and (256.5) it follows that the mass number during -decay decreases by 4, but does not change during -decay. Therefore, for all nuclei of the same radioactive family, the remainder when dividing the mass number by 4 is the same. Thus, there are four different radioactive families, for each of which the mass numbers are given by one of the following formulas:
A = 4n, 4n+1, 4n+2, 4n+3,
Where n- whole positive number. Families are named by the longest-lived (with the longest half-life) “ancestor”: the families of thorium (from), neptunium (from), uranium (from) and sea anemone (from). The final nuclides, respectively, are , , , , i.e. the only family of neptunium (artificially radioactive nuclei) ends with a nuclide Bi, and all the rest (naturally radioactive nuclei) are nuclides Pb.
§ 257. Laws of decay
Currently, more than two hundred -active nuclei, mainly heavy ( A > 200, Z> 82). Only a small group of -active nuclei occur in areas with A= 140 ¸ 160 (rare earths). -Decomposition obeys the displacement rule (256.4). An example of -decay is the decay of an isotope of uranium with the formation Th:
The velocities of particles emitted during decay are very high and range for different nuclei from 1.4 × 10 7 to 2 × 10 7 m/s, which corresponds to energies from 4 to 8.8 MeV. According to modern ideas, -particles are formed at the moment of radioactive decay when two protons and two neutrons moving inside the nucleus meet.
Particles emitted by a specific nucleus usually have a certain energy. More subtle measurements, however, have shown that the energy spectrum of -particles emitted by a given radioactive element exhibits a “fine structure”, that is, several groups of -particles are emitted, and within each group their energies are practically constant. The discrete spectrum of -particles indicates that atomic nuclei have discrete energy levels.
-decay is characterized by a strong relationship between half-life and energy E flying particles. This relationship is determined empirically Geiger-Nattall law(1912) (D. Nattall (1890-1958) - English physicist, H. Geiger (1882-1945) - German physicist), which is usually expressed as a connection between mileage(the distance traveled by a particle in a substance before it comes to a complete stop) - particles in the air and the radioactive decay constant:
(257.1)
Where A And IN- empirical constants, . According to (257.1), the shorter the half-life of a radioactive element, the greater the range, and therefore the energy of the particles emitted by it. The range of particles in the air (at normal conditions) is several centimeters; in denser media it is much smaller, amounting to hundredths of a millimeter (-particles can be detained with an ordinary sheet of paper).
Rutherford's experiments on the scattering of -particles on uranium nuclei showed that -particles up to an energy of 8.8 MeV experience Rutherford scattering on nuclei, i.e., the forces acting on -particles from the nuclei are described by Coulomb's law. This type of scattering of -particles indicates that they have not yet entered the range of action nuclear forces, i.e. we can conclude that the nucleus is surrounded by a potential barrier, the height of which is not less than 8.8 MeV. On the other hand, -particles emitted by uranium have an energy of 4.2 MeV. Consequently, -particles fly out from the -radioactive nucleus with an energy noticeably lower than the height of the potential barrier. Classical mechanics could not explain this result.
An explanation for -decay is given by quantum mechanics, according to which the escape of an -particle from the nucleus is possible due to the tunnel effect (see §221) - the penetration of an -particle through a potential barrier. There is always a non-zero probability that a particle with an energy less than the height of the potential barrier will pass through it, i.e., indeed, particles can fly out of a radioactive nucleus with an energy less than the height of the potential barrier. This effect is entirely due to the wave nature of -particles.
The probability of a particle passing through a potential barrier is determined by its shape and is calculated based on the Schrödinger equation. In the simplest case of a potential barrier with rectangular vertical walls (see Fig. 298, A) the transparency coefficient, which determines the probability of passing through it, is determined by the previously discussed formula (221.7):
Analyzing this expression, we see that the transparency coefficient D the longer (therefore, the shorter the half-life) the smaller in height ( U) and width ( l) the barrier is in the path of the -particle. In addition, with the same potential curve, the greater the energy of a particle, the smaller the barrier to its path. E. Thus, the Geiger-Nattall law is qualitatively confirmed (see (257.1)).
§ 258. -Disintegration. Neutrino
The phenomenon of -decay (in the future it will be shown that there is and (-decay) obeys the displacement rule (256.5)
and is associated with the release of an electron. I had to overcome a whole series difficulties with the interpretation of -decay.
First, it was necessary to substantiate the origin of the electrons emitted during the decay process. The proton-neutron structure of the nucleus excludes the possibility of an electron escaping from the nucleus, since there are no electrons in the nucleus. The assumption is that electrons fly out not from the nucleus, but from electron shell, is untenable, since then optical or x-ray radiation, which is not confirmed by experiments.
Secondly, it was necessary to explain the continuity of the energy spectrum of emitted electrons (the energy distribution curve of -particles typical for all isotopes is shown in Fig. 343).
How can active nuclei, which have well-defined energies before and after decay, eject electrons with energy values from zero to a certain maximum? That is, the energy spectrum of emitted electrons is continuous? The hypothesis that during -decay electrons leave the nucleus with strictly defined energies, but as a result of some secondary interactions they lose one or another share of their energy, so that their original discrete spectrum turns into a continuous one, was refuted by direct calorimetric experiments. Since the maximum energy is determined by the difference in the masses of the mother and daughter nuclei, then decays in which the electron energy< , как бы протекают с нарушением закона сохранения энергии. Н. Бор даже пытался обосновать это нарушение, высказывая предположение, что закон сохранения энергии носит статистический характер и выполняется лишь в среднем для large number elementary processes. This shows how fundamentally important it was to resolve this difficulty.
Thirdly, it was necessary to deal with spin non-conservation during -decay. During -decay, the number of nucleons in the nucleus does not change (since the mass number does not change A), therefore the spin of the nucleus, which is equal to an integer for even A and half-integer for odd A. However, the release of an electron with spin /2 should change the spin of the nucleus by the amount /2.
The last two difficulties led W. Pauli to the hypothesis (1931) that during -decay, another neutral particle is emitted along with the electron - neutrino. The neutrino has zero charge, spin /2 and zero (or rather< 10 -4 ) массу покоя; обозначается . Впоследствии оказалось, что при - decay, it is not neutrinos that are emitted, but antineutrino(antiparticle in relation to neutrinos; denoted by ).
The hypothesis of the existence of neutrinos allowed E. Fermi to create the theory of -decay (1934), which has largely retained its significance to this day, although the existence of neutrinos was experimentally proven more than 20 years later (1956). Such a long “search” for neutrinos is associated with great difficulties due to the lack of electrical charge and mass in neutrinos. Neutrino is the only particle that does not participate in either strong or electromagnetic interactions; the only type of interaction in which neutrinos can take part is weak interaction. Therefore, direct observation of neutrinos is very difficult. The ionizing ability of neutrinos is so low that one ionization event in the air occurs per 500 km of travel. The penetrating ability of neutrinos is so enormous (the range of neutrinos with an energy of 1 MeV in lead is about 1018 m!), which makes it difficult to contain these particles in devices.
For the experimental detection of neutrinos (antineutrinos), an indirect method was therefore used, based on the fact that in reactions (including those involving neutrinos) the law of conservation of momentum is satisfied. Thus, neutrinos were discovered by studying the recoil of atomic nuclei during -decay. If during the decay of a nucleus an antineutrino is ejected along with an electron, then the vector sum of three impulses - the recoil nucleus, the electron and the antineutrino - should be equal to zero. This has indeed been confirmed by experience. Direct detection neutrinos became possible only much later, after the advent of powerful reactors that made it possible to obtain intense neutrino fluxes.
The introduction of neutrinos (antineutrinos) made it possible not only to explain the apparent non-conservation of spin, but also to understand the issue of continuity of the energy spectrum of ejected electrons. The continuous spectrum of -particles is due to the distribution of energy between electrons and antineutrinos, and the sum of the energies of both particles is equal to . In some decay events, the antineutrino receives more energy, in others - the electron; at the boundary point of the curve in Fig. 343, where the electron energy is equal to , all the decay energy is carried away by the electron, and the antineutrino energy is zero.
Finally, let us consider the question of the origin of electrons during -decay. Since the electron does not fly out of the nucleus and does not escape from the shell of the atom, it was assumed that the electron is born as a result of processes occurring inside the nucleus. Since during -decay the number of nucleons in the nucleus does not change, a Z increases by one (see (256.5)), then the only possibility of simultaneous implementation of these conditions is the transformation of one of the neutrons - the active nucleus into a proton with the simultaneous formation of an electron and the emission of an antineutrino:
(258.1)
This process is accompanied by the fulfillment of conservation laws electric charges, momentum and mass numbers. In addition, this transformation is energetically possible, since the rest mass of the neutron exceeds the mass of the hydrogen atom, i.e., the proton and electron combined. This difference in mass corresponds to an energy equal to 0.782 MeV. Due to this energy, spontaneous transformation of a neutron into a proton can occur; energy is distributed between the electron and the antineutrino.
If the transformation of a neutron into a proton is energetically favorable and generally possible, then radioactive decay of free neutrons (i.e., neutrons outside the nucleus) should be observed. The discovery of this phenomenon would be a confirmation of the stated theory of decay. Indeed, in 1950, in high-intensity neutron fluxes arising in nuclear reactors, the radioactive decay of free neutrons was discovered, occurring according to scheme (258.1). The energy spectrum of the resulting electrons corresponded to that shown in Fig. 343, and the upper limit of the electron energy turned out to be equal to that calculated above (0.782 MeV).
Radioactive decay of atomic nuclei occurs spontaneously and leads to a continuous decrease in the number of atoms of the original radioactive isotope and the accumulation of atoms of the decay product.
The rate at which radionuclides decay is determined only by the degree of instability of their nuclei and is independent of any factors that usually influence the rate of physical and chemical processes(pressure, temperature, chemical form of a substance, etc.). The decay of each individual atom is a completely random event, probabilistic and independent of the behavior of other nuclei. However, if there is a sufficiently large number of radioactive atoms in the system, general pattern, which consists in the fact that the number of atoms of a given radioactive isotope decaying per unit time always constitutes a certain fraction characteristic of a given isotope of the total number of atoms that have not yet decayed. The number of DUU atoms that have undergone decay in a short period of time D/ is proportional to the total number of undecayed radioactive atoms DU and the value of the DL interval. This law can be mathematically represented as the ratio:
-AN = X ? N? D/.
The minus sign indicates that the number of radioactive atoms N decreases. Proportionality factor X is called decay constant and is a constant characteristic of a given radioactive isotope. The law of radioactive decay is usually written as a differential equation:
So, law of radioactive decay can be formulated as follows: per unit time, the same part of the available nuclei of a radioactive substance always decays.
Decay constant X has the dimension of inverse time (1/s or s -1). The more X, the faster the decay of radioactive atoms occurs, i.e. X characterizes the relative decay rate for each radioactive isotope or decay probability atomic nucleus in 1 s. The decay constant is the fraction of atoms decaying per unit time, an indicator of the instability of a radionuclide.
Value - absolute rate of radioactive decay -
called activity. Radionuclide activity (A) - This is the number of atomic decays occurring per unit time. It depends on the number of radioactive atoms at a given time (AND) and on the degree of their instability:
A=Y ( X.
The SI unit of activity is becquerel(Bq); 1 Bq - activity at which one nuclear transformation occurs per second, regardless of the type of decay. Sometimes an off-system unit of measurement of activity is used - the curie (Ci): 1Ci = 3.7-10 10 Bq (the number of decays of atoms in 1 g 226 RAA in 1 s).
Since activity depends on the number of radioactive atoms, this value serves as a quantitative measure of the content of radionuclides in the sample being studied.
In practice, it is more convenient to use the integral form of the law of radioactive decay, which has the following form:
where УУ 0 - number of radioactive atoms at the initial moment of time / = 0; - the number of radioactive atoms remaining at the moment
time /; X- decay constant.
To characterize radioactive decay, often instead of a decay constant X They use another quantity derived from it - the half-life. Half-life (T]/2)- this is the period of time during which half of the initial number of radioactive atoms decays.
Substituting the values G = into the law of radioactive decay T 1/2 And AND (= Af/2, we get:
VU 0 /2 = # 0 e~ xt og-
1 /2 = e~ xt "/2 -, A e xt "/ 2 = 2 or HT 1/2 = 1p2.
The half-life and decay constant are related by the following relationship:
T x/2=1п2 А = 0.693 /X.
Using this relationship, the law of radioactive decay can be presented in another form:
TU, = УУ 0 e Apg, "t t
N = And 0? e-°’ t - ( / t 02.
From this formula it follows that the longer the half-life, the slower the radioactive decay occurs. Half-lives characterize the degree of stability of the radioactive nucleus and vary widely for different isotopes - from fractions of a second to billions of years (see appendices). Depending on their half-life, radionuclides are conventionally divided into long-lived and short-lived.
The half-life, along with the type of decay and energy of radiation, is the most important characteristic any radionuclide.
In Fig. Figure 3.12 shows the decay curve of a radioactive isotope. The horizontal axis represents time (in half-lives), and the vertical axis- the number of radioactive atoms (or activity, since it is proportional to the number of radioactive atoms).
The curve is exponent and asymptotically approaches the time axis without ever crossing it. After a period of time equal to one half-life (Г 1/2), the number of radioactive atoms decreases by 2 times; after two half-lives (2Г 1/2), the number of remaining atoms again decreases by half, i.e. 4 times from their initial number, after 3 7" 1/2 - 8 times, after
4G 1/2 - 16 times, through T half-lives Г ]/2 - in 2 t once.
Theoretically, the population of atoms with unstable nuclei will decrease to infinity. However, from a practical point of view, a certain limit should be designated when all radioactive nuclides have decayed. It is believed that this requires a period of time of 107^, 2, after which less than 0.1% of radioactive atoms will remain of the original amount. Thus, if we take into account only physical decay, for complete cleansing biosphere from 90 Bg (= 29 years) and |37 Sz (T|/ 2 = 30 years) of Chernobyl origin will require 290 and 300 years, respectively.
Radioactive balance. If, during the decay of a radioactive isotope (parent), a new radioactive isotope (daughter) is formed, then they are said to be genetically related to each other and form radioactive family(row).
Let us consider the case of genetically related radionuclides, of which the parent is long-lived and the daughter is short-lived. An example is strontium 90 5g, which is converted by (3-decay ( T /2 = 64 h) and turns into a stable zirconium nuclide ^Ъх(see Fig. 3.7). Since 90 U decays much faster than 90 5g, after some time there will come a moment when the amount of decaying 90 8g at any moment will be equal to the amount of decaying 90 U. In other words, the activity of the parent 90 8g (D,) will be equal to the activity of the daughter 90 U (L 2). When this happens, 90 V is considered to be in secular equilibrium with its parent radionuclide 90 8g. In this case the relation holds:
A 1 = L 2 or X 1? = X 2?УУ 2 or: Г 1/2(1) = УУ 2: Г 1/2(2) .
From the above relationship it follows that than more likely radionuclide decay (To) and, accordingly, a shorter half-life (T ]/2), the less its atoms are contained in a mixture of two isotopes (AO-
Establishing such equilibrium requires a time of approximately 7T ]/2 daughter radionuclide. Under conditions of secular equilibrium, the total activity of a mixture of nuclides is twice as high as the activity of the parent nuclide at a given point in time. For example, if at the initial time the drug contains only 90 8g, then after 7T/2 the longest-lived member of the family (except for the ancestor of the series), a secular equilibrium is established, and the decay rates of all members of the radioactive family become the same. Considering that the half-lives for each member of the family are different, the relative amounts (including mass) of nuclides in equilibrium are also different. The less T )
